We propose a statistical model defined on tetravalent three-dimensional lattices in general and the three-dimensional diamond network in particular where the splitting of randomly selected nodes leads to a spatially disordered network, with decreasing degree of connectivity. The terminal state, that is reached when all nodes have been split, is a dense configuration of self-avoiding walks on the diamond network. Starting from the crystallographic diamond network, each of the four-coordinated nodes is replaced with probability *p* by a pair of two edges, each connecting a pair of the adjacent vertices. For all values 0 ≤ *p ≤* 1 the network percolates, yet the fraction *f*_{p} of the system that belongs to a percolating cluster drops sharply at *p*_{c} = 1 to a finite value *f*_{p}^{c}. This transition is reminiscent of a percolation transition yet with distinct differences to standard percolation behaviour, including a finite mass *f*_{p}^{c} > 0 of the percolating clusters at the critical point. Application of finite size scaling approach for standard percolation yields scaling exponents for *p →* *p*_{c} that are different from the critical exponents of the second-order phase transition of standard percolation models. This transition significantly affects the mechanical properties of linear-elastic realizations (e.g. as custom-fabricated models for artificial bone scaffolds), obtained by replacing edges with solid circular struts to give an effective density *ϕ*. Finite element methods demonstrate that, as a low-density cellular structure, the bulk modulus *K* shows a cross-over from a compression-dominated behaviour, K(*ϕ* ) α *ϕ*^{К} with *К* ≈ 1, at *p* = 0 to a bending-dominated behaviour with *К* ≈ 2 at *p* = 1.